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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::DiagonalsAlternating \-\- points in diagonal stripes of alternating directions .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::DiagonalsAlternating; \& my $path = Math::PlanePath::DiagonalsAlternating\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path follows successive diagonals going from the Y axis down to the X axis and then back again, .PP .Vb 10 \& 7 | 29 \& 6 | 28 30 \& 5 | 16 27 31 \& 4 | 15 17 26 ... \& 3 | 7 14 18 25 \& 2 | 6 8 13 19 24 \& 1 | 2 5 9 12 20 23 \& Y=0 | 1 3 4 10 11 21 22 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 5 6 .Ve .PP The triangular numbers 1,3,6,10,etc k*(k+1)/2 are the start of each run up or down alternately on the X axis and Y axis. N=1,6,15,28,etc on the Y axis (Y even) are the hexagonal numbers j*(2j\-1). N=3,10,21,36,etc on the X axis (X odd) are the hexagonal numbers of the second kind j*(2j+1). .IX Xref "Triangular numbers Hexagonal numbers" .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pattern. For example to start at 0, .PP .Vb 1 \& n_start => 0 \& \& 4 | 14 \& 3 | 6 13 \& 2 | 5 7 12 \& 1 | 1 4 8 11 \& Y=0 | 0 2 3 9 10 \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& X=0 1 2 3 4 .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::DiagonalsAlternating\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::DiagonalsAlternating\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::DiagonalsAlternating->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::DiagonalsAlternating\->new (n_start => $n)""" 4 .el .IP "\f(CW$path = Math::PlanePath::DiagonalsAlternating\->new (n_start => $n)\fR" 4 .IX Item "$path = Math::PlanePath::DiagonalsAlternating->new (n_start => $n)" .PD Create and return a new path object. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. .Sp For \f(CW\*(C`$n < 1\*(C'\fR the return is an empty list, it being considered the path begins at 1. .SH "FORMULAS" .IX Header "FORMULAS" .SS "Rectangle to N Range" .IX Subsection "Rectangle to N Range" Within each row increasing X is increasing N, and in each column increasing Y is increasing N. So in a rectangle the lower left corner is the minimum N and the upper right is the maximum N. .PP .Vb 8 \& | N max \& | \-\-\-\-\-\-\-\-\-\-+ \& | | ^ | \& | | | | \& | | \-\-\-\-> | \& | +\-\-\-\-\-\-\-\-\-\- \& | N min \& +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- .Ve .SH "OEIS" .IX Header "OEIS" Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include .Sp .RS 4 (etc) .RE .PP .Vb 5 \& n_start=1 \& A131179 N on X axis (extra initial 0) \& A128918 N on Y axis (extra initial 1) \& A001844 N on X=Y diagonal \& A038722 permutation N at transpose Y,X \& \& n_start=0 \& A003056 X+Y \& A004247 X*Y \& A049581 abs(X\-Y) \& A048147 X^2+Y^2 \& A004198 X bit\-and Y \& A003986 X bit\-or Y \& A003987 X bit\-xor Y \& A004197 min(X,Y) \& A003984 max(X,Y) \& A101080 HammingDist(X,Y) \& A023531 dSum = dX+dY, being 1 at N=triangular+1 (and 0) \& A046092 N on X=Y diagonal \& A061579 permutation N at transpose Y,X \& \& A056011 permutation N at points by Diagonals,direction=up order \& A056023 permutation N at points by Diagonals,direction=down \& runs alternately up and down, both are self\-inverse .Ve .PP The coordinates such as A003056 X+Y are the same here as in the Diagonals path. \f(CW\*(C`DiagonalsAlternating\*(C'\fR transposes X,Y \-> Y,X in every second diagonal but forms such as X+Y are unchanged by swapping to Y+X. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsOctant .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .